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For complex numbers z and w , if |z|^2 w-|w|^2 z=z-w . Which of the following are true :

Options

  1. Az=w
  2. Bz w =1
  3. Cz=w+2
  4. Dz w=1

Correct answer

D. z w=1

Step-by-step solution

Given that z and are two complex numbers. To prove |z|^2 w-|w|^2 z=z-w z=w or z w =1 First let us consider |z|^2 w-|w|^2 z=z-w .....(1) aligned & z (1+|w|^2 )=w (1+|z|^2 ) & z w = 1+|z|^2 1+|w|^2 & ( z w ) = z w z w = z w aligned z w=z w .....(2) Again from Eq. (1), aligned & z z w-w w z=z-w & z( z w-1)-w( w z-1)=0 & z(z w -1)-w(z w -1)=0 [Using Eq. (2)] & (z w -1)(z-w)=0 & z w =1 or z=w aligned

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